The present invention relates to target location methods and systems and navigational methods and systems, and more particularly to the determination of the precise location of an object in relationship to two viewing locations and to the precise location of an observation platform in relationship to a ground station.
At the present time there are a number of systems and methods utilized to relate the coordinate system at one location to the coordinate system at a different location. For example, if one location is an aircraft and the second location is a ground station, there is a need to relate the coordinate system of the aircraft, for example, for purposes of navigation, to the coordinate system of the ground station. The relationship between the coordinate system of the aircraft and the ground may be indicated by an "inertial platform" which is a system of two or three gyroscopes whose spin axes are mutually perpendicular.
Another example arises in target acquisition systems. The target is seen or sensed by an observer or sensor on an observation platform, such as a helicopter. One of the major problems involved in using such an observation platform is to translate the range, azimuth and elevation of the observed or sensed target from the reference frame (coordinate system) of the platform into the location of the target in the reference frame of the ground station. In general, the coordinate system of the observation platform will not be exactly aligned with that of the ground control station.
Traditionally, the relationship between the two coordinate systems may be determined by a method known as "Euler angles." A further description of the Euler angle method is set forth below. In general, however, the difficulty arises not because of mathematical inaccuracies in that system of transformation, but rather because of errors introduced by the physical system which attempts to measure angles corresponding to the mathematical definitions of the Euler angles. In the Euler angle method the rotated coordinate system is obtained from the unrotated coordinate system by a first angular motion about one axis followed by a second angular motion about a locally perpendicular second axis, and then followed by a third angular motion about the now displaced local axis corresponding to the first axis. By the third angular motion, the axes have been skewed so that it is difficult to obtain angles corresponding to the defined Euler angles with great accuracy in the physical system.
The angles measured by the inertial system are attempted to be as close as possible to the definitions of the Euler angles set forth below. The measured angles are then used as the basis for the standard Euler angle rotation matrix in order to define the coordinate system of the aircraft. This is subject to several problems: (1) drift in the gyroscope; (2) lack of sufficiently accurate angular pickoffs of reasonable size; and (3) the angles as actually measured are close to but not exactly the Euler angles as defined in the textbooks due to limitations in the measuring devices. There have been various proposals for physical systems involving electromechanical take-offs and axes which are rotated in different directions to compensate for errors in measurement. However, none of these has fully solved the problem of accuracy.
"Euler angles" are the three angular parameters that may be used to specify the orientation of a body with respect to reference axes. The definition generally given is as follows:
OXYZ is a right-handed Cartesian (orthagonal) set of fixed coordinate axes and Oxyz is a set attached to the rotating body.
The orientation of Oxyz can be produced by three successive rotations about the fixed axes starting with Oxyz parallel to OXYZ. Rotate through (1) the angle .epsilon..sub.1 counterclockwise about OZ (OZ = O.sub.z); (2) the angle .epsilon..sub.2 counterclockwise about Ox (O.sub.x .noteq. OX); and (3) the angle .epsilon..sub.3 counterclockwise about O.sub.z (O.sub.z .noteq. OZ) again. The line of intersection OK of the xy and XY planes is called the line of nodes.
A rotation about OZ is denoted, for example, by Z (angle). Then the complete rotation is, symbolically EQU R (.epsilon..sub.1, .epsilon..sub.2, .epsilon..sub.3) = Z (.phi.)x(.theta.)z(.epsilon..sub.1)
where the rightmost operation is done first.
A point P will have coordinates (x,y,z) with respect to the body axes and (X,Y,Z) with respect to the fixed ones. These are related by the linear equations: EQU x = X cos (x,X) + Y cos (x,Y) + Z cos (x,Z) EQU y = X cos (y,X) + Y cos (y,Y) + Z cos (y,Z) EQU z = X cos (z,X) + Y cos (z,Y) + Z cos (z,Z)
where (x,X) is the angle between the axes Ox and OX, etc. The nine direction cosines are expressed in terms of the three Euler angles.
It is apparent that no operation in R (.epsilon..sub.1, .epsilon..sub.2, .epsilon..sub.3) can be replaced by a combination of the other two. Therefore, three parameters are needed to specify the orientation, and the amounts of the angles are unique (barring addition 360.degree. rotations). In dynamical problems of rotating bodies, .epsilon..sub.1, .epsilon..sub.2 and .epsilon..sub.3 can be used as independent angular coordinates.